Monday, January 21, 2019

Two easy projects

Sometimes it's fun just to make simple projects.  Here are two I built this weekend.  A wooden yo-yo

and a sundial for my office.

The sundial is intended to be mounted on the wall, which does not lie in a cardinal direction.  It's therefore what is called "vertical declining" sundial.  My office wall is parallel to 130 degrees, so the gnomon lies off center and the spacing of the hour lines isn't uniform.  I built it according to the description given in

A. Waugh, Sundials: Their Theory and Construction, Dover, 1973.

Hopefully it'll work!

Thursday, January 10, 2019

Clock 4 current power consumption

Continuing the thoughts from the previous post... How much power does the current Clock 4 pendulum and count wheel consume?  Especially, how much weight is really necessary to drive it?

I'll treat the pendulum rod and bob as two separate weights...

Rod = 28.86 oz = 0.818 kg, centered at 24" = 0.61 m
Bob = 26.75 oz = 0.758 kg, centered at 45" = 1.14 m

Potential energy for a swinging weight = m g L (1-cos(angle))

The amount of energy at the top of the test swing (4.8 degrees) is

( 0.818 kg * 0.61 m + 0.758 kg * 1.14 m ) * 9.8 N/kg * ( 1 - cos (4.8 degrees) ) = 0.046850 J

At the bottom of the test swing (2.4 degrees), the energy is

( 0.818 kg * 0.61 m + 0.758 kg * 1.14 m ) * 9.8 N/kg * ( 1 - cos (2.4 degrees) ) = 0.011718 J.

Assuming one period of the pendulum is 2 seconds (it's not, but will eventually be):
  • The unloaded pendulum takes 65 periods to consume that energy = 0.270 mW
  • The pendulum driving the pulling pallet consumes this energy in 54 periods = 0.325 mW
  • The complete count wheel assembly consumes this energy in 50 periods = 0.351 mW
We can conclude that
  • The count wheel assembly consumes 0.081 mW,
  • of which 0.026 mW is due to the backstop.
These power figures are somewhat in line with my previous clocks.  Clock 1 runs on 0.5 mW and Clock 3 runs on 0.8 mW.  So thus far, Clock 3 is more efficient by a bit.

Assume that the escapement is triggered once per minute, is geared through a 10:1 gear mesh, and is driven by a 1" diameter barrel.  How much weight is required for all of these power requirements?

The weight falls at an average speed of pi * 0.0254 m / (36000 s) = 2.216e-6 m/s.

Thus, it takes
  • 12.4 kg = 27.4 lb to drive the unloaded pendulum,
  • 3.7 kg = 8.2 lb to drive the count wheel (without the pendulum), and
  • 16.1 kg = 35.5 lb to drive the pendulum and count wheel assembly.
Way too high, I think!  I need to either improve the pendulum's Q or scrap the idea of the 10:1 gear mesh.

For testing purposes, if I were to drive the clock from the pin escape wheel directly, which has a 3/4" pinion, the weight falls at an average speed of pi * 0.75 in * 0.0254 m/in / (3600 s) = 1.6624e-05 m/s.  The amount of weight necessary to drive the pendulum and count wheel assembly becomes 2.16 kg = 4.8 lb.  (This may not be entirely safe since the pin escape wheel arbor isn't very strong.)

Wednesday, January 9, 2019

Clock 4 pendulum measurements

Here are some measurements of the clock 4 pendulum, trying to get a handle on its performance issues.  Woodward is adamant that limiting count wheel friction was major concern in his designs.  He employed a number of countermeasures, including anti-friction rollers, a polished acrylic count wheel, lightweight stainless steel pallets, and the merest hint of watch oil.  I don't know that my situation calls for such measures, but I figured I ought to investigate.

The pendulum is a solid square black walnut rod about 2" on a side, and is 48" from knife edge to bottom.  It weighs 28.86 oz, which is fairly uniformly distributed along its length.  The pendulum was fitted with a crude bob constructed of short copper-clad steel rods bound together with a rubber band located 45" (on center) from the knife edge, weighing 26.75 oz.

I measured pendulum amplitudes as deflections from equilibrium. 

Provided the amplitude is greater than 2.5" (3 degrees) and less than 5" (6 degrees), the count wheel advances reliably.  The count wheel does not advance at all when the amplitude is less than 2.25" (2.6 degrees). Double counting occurs when the amplitude is greater than 5.5" (6.6 degrees).

Here are counts of pendulum full periods starting at 4" (4.8 degrees) and ending at 2" (2.4 degrees), which is basically a half-time.  Pendulum Q can be estimated from this by Q = 4.532 * number of periods to halve the amplitude.
  • Unloaded pendulum: 65, 72, 68.  Median Q = 308
  • Pendulum driving pull pallet and count wheel, but no backstop: 58, 54, 54.  Median Q = 245
  • Pendulum driving count wheel normally: 52, 45, 50.  Median Q = 226.
This indicates a count wheel-only reliable run time of about 100 seconds, which I've confirmed approximately on previous days.  If you push it a bit, you can sometimes do better on occasion.

There definitely is a noticeable change in loaded Q caused by driving the count wheel, as Woodward warns.  But, the unloaded Q figures are probably the source of my trouble, though.  The unloaded Q is around the same as a marine chronometer's balance (and not a good one at that), and that needs an impulse every period to keep running!  (I already know that the clock can run if it impulses every second.... it's not supposed to do that, though!)

Although this is probably excessive for my needs, Woodward has a table that lists a "heavy seconds pendulum" at Q = 15 000.  I think I need a better resonator!

Triggering the escapement certainly consumes energy, possibly a large amount of energy.  But it's unclear how exactly to measure that accurately...

Monday, January 7, 2019

Clock 4 escapement triggers

The next step of constructing Clock 4 is the intermittent triggering mechanism.  Once per minute (one rotation of the count wheel), it triggers the impulse hook to grab one pin of the pin escape wheel.

The impulse hook was cut from the pendulum rod (mostly for aesthetics).  This version has a brass hook rooted in the block and anchored with super glue.  I later replaced this with a stiffer steel one.

The hook is counterweighted by filling the wooden block with lead.  This was sufficient for the brass hook, but not for the steel hook.  I added a screw and nut outrigger counterweight for that.  The steel hook turned out to be a good idea because the brass one was really very pliant, and was getting distorted by each impulse. 

The hook assembly rides on a brass pin on the pendulum.

The impulse hook is triggered by an assembly that sits behind the count wheel.  The straight segment gets grabbed by the count wheel driving pallet (hook) once a minute, and pushes the flat segment against the impulse hook to engage it.

I also made a wood and brass key to wind the clock.

Here is a video the escapement being triggered successfully from the count wheel.  (Hemostats are useful to keep parts in place...)

This has taken the past two days to get it adjusted.  Here is a video of an amusing -- and vexing -- fail mechanism.  Watch to the end... it gets worse!

Next up: the pendulum is indeed not running long enough (as the previous post probably suggests...).  I suspect I do need to increase the weight of the pendulum bob, regardless of the power needs, and figure out how to reduce the friction.  That first requires finding where the friction is...

Sunday, January 6, 2019

Should you add more weight to the pendulum bob when the clock doesn't run?

Short answer: no!

Medium answer: adding more weight to the bob increases the per-period energy requirement of the clock, but only up to a limit.  So once you have the clock running reliably, you can (and should) add more weight to the bob to improve its timekeeping stability.

When clock doesn't run because not enough energy is getting refreshed into the pendulum (or other resonator), it's tempting to look for easy fixes.  Adding more weight to the pendulum bob certainly delays the inevitable, since the clock will run longer before stopping.  But it has a certain futile feel to it... will you ever add enough so that it will never stop?  Sadly, you cannot.  Fixing the frictional losses (best) or increasing the drive power are the only solutions.

The reason is a straightforward derivation ending in a simple formula.

First of all, let the angular deflection of the pendulum be A = A(t), a function of time t.  For small angles, this is governed by

A'' + (c/m) A' + (g/L) A = 0,

where m is the pendulum bob mass, g is the acceleration due to gravity, L is the length of the pendulum, and c is the frictional loss constant.  By the usual process for solving such a differential equation, the envelope of the oscillations will naturally decay like

A(t) = exp( - ct/(2m) ) A(0) ( .. trig functions .. ).

If T is the time of one period, the max amplitude is given by

A(T) = exp( -cT/(2m) ) A(0).

Now switching to discuss energy, the height of the pendulum is found by a little geometry...

... to be given by

h = L(1-cos A(t)).

Therefore, the energy change from one period to the next is

dE = mgL(cos(A(T)) - cos(A(0))) = mgL(cos(exp( -cT/(2m) ) A(0)) - cos(A(0)))

again applying a small angle approximation,

This expression is the one we're after, and really we want to know how it changes as we change m.  Clearly at m = 0, the change in energy is zero. Taylor expanding in m, we have the behavior for large m is approximately

Here is a plot of the overall behavior

The takeaway is that as you increase the mass of the pendulum bob, you must supply more energy to sustain oscillations, but only up to a limit.  For stable, reliable operation, you should ensure that the drive supplies at least that limiting amount of energy first first, before increasing bob weight.  Minimizing frictional losses should be the first priority -- which decreases c -- before trying to increase drive weight.  Only after the clock runs reliably should you attempt to increase the pendulum bob weight.

Tuesday, January 1, 2019

Clock 4 detent works!

Remaking the detent a few times did the trick.  Each time it worked a little better than the one before, as I flushed out the bugs.  I had a scare where I damaged the gate, but a little super glue seems to be holding it together.  Here is the detent that finally does the job.

The detent properly releases one pin at a time when recoiled by hand with a weight of just about 2.2 lb on the great wheel.

I did not have to modify the pin wheel. It is helpful to have a banking so the detent is held in a convenient position if all the weight is removed.  This also gave a good opportunity for testing the winding mechanism, which does indeed work.

Monday, December 31, 2018

Clock 4 detent issues

I installed a new(er) 1/3 hp 1725 rpm motor on my lathe since the bearings on the old one were dead.  It runs much better than before!

The detent mechanism for Clock 4 uses a gate invented by Philip Woodward (I think).  The detent sits on a pivot near the pin escape wheel.

The detent is fairly long, but just press fit into the frame.

The detent is cut from a small piece of white oak.

Here is the detent after shaping.

There are many issues with the detent, and it doesn't run at the moment:
  • The gate is very thin.  I broke two detents already
  • Woodward didn't seem to bank his detent, but it looks like I need to since wood has more flexibility than metal
  • The catch for holding the pin is very touchy as to how deep it is.  Woodward suggests that it might work as just a small depression, but this caused the pins to jump out.  Too deep, and they can't clear when the escape wheel recoiled... in which case the pins stick.
  • The pins of the escape wheel are too inaccurate in their placement
  • The pins of the escape wheel are too inaccurate in their vertical alignment
  • The pins of the escape wheel are not all the same diameter (because some of them split in the process of being installed).
  • The relative positioning of the catch and the gate slot is quite delicate, and there isn't much clearance.
  • The counterweight portion of the detent governs how much weight is needed to run the escapement.  This needs to be very light.
A few times, I could feel the escapement "almost working" under my hand, but it wasn't consistent enough to run under a weight.